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Geometric Series

It's the last section, Shmoopers. Geometric series are just the added together version of a geometric sequence. We use the same sigma notation we used with arithmetic series so we have a general form that looks like this:

In this series, our numbers will start when n = 1 and go all the way to infinity. Geometric series are unique in this way. Not only can we find partial sums like we did with arithmetic sequences, we can find the overall sum as well. We will do both; cool your jets.

Before we jump into sample problems we'll need two formulas to find these sums. The first is the formula for the sum of an infinite geometric series.

This formula really couldn't be much simpler. All we need is the first term and the common ratio and boom—we have the sum.

Unfortunately, and this is a big unfortunately, this formula will only work when we have what is known as a convergent geometric series. A convergent series is one whose partial sums get closer to a certain value as the number of terms increases. We think this will help…

The series with the terms 1, 2, 4, 8, 16, 32,… does not converge because every time we put another number into your sum, the sum gets a lot bigger. We can see that each time we add in another number, the sum is going to get larger and larger and larger and larger and larger and larger…you get the idea. This series is divergent.

On the other hand, the series with the terms  has a sum that also increases with each additional term. However, each time we add in another term, the sum is not going to get that much bigger. This is especially true when we add in terms like  . This is only the 20th term of this series but it is very small.

While the ideas of convergence and divergence are a little more involved than this, for now, this working knowledge will do. In fact, we can tell if an infinite geometric series converges based simply on the r value. When |r| < 1 the series converges. When |r| ≥ 1 the series diverges.

This means it only makes sense to find sums for the convergent series since divergent ones have sums that are infinitely large. This is true even though the formula we gave you technically gives you a number when you put in a1 and r.

Sample Problem

Find the sum of the infinite geometric series given by: .

Solution: Nice try. In this series r = 2. 2 > 1. That means the series diverges and its sum is infinitely large.

Sample Problem

Find the sum of the infinite geometric series given by: .

Solution: Much better. We will use our formula and then get on with our lives.

Remember how awhile back we said there were two formulas we'll need? Well, we did. And here it is. The second formula you need is the one for the nth partial sum. Remember we call that Sn. It simply means that we are only going to add up the first n terms. That might be 5, 10, or 20 terms. The bottom line is we're not adding them all the way to infinity.

This formula is really close to our original formula. The only difference is the (1 – rn). It works for any geometric series regardless of the value of r.

Sample Problem

Find the fifth partial sum of the geometric series given by: .

Solution: Let's take the partial sum formula and substitute a1 and r. We can also put in 5 for n since we're asked to find the fifth partial sum.

Notice our answer here is big. But that is kind of expected since this series would be considered a divergent infinite geometric series and, as a result, have some seriously big partial sums along the way.

Sample Problem

Find the sum:  .

Solution: Formula to the rescue…again. Note that n will be 6 in our formula since our series is set up to go from n = 1 to n = 6.

It sure isn't pretty. But it works.

Nice job. That's a wrap.

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